Mathematics — Grade 5

The child represents simple counting tasks and solves them. They arrange the combinations so that you can see whether all of them are there, for example as a list or a table. The curriculum refers to point VS3.1.2 from the primary school (Volksschule) for this. It does not give a specific number range. Point U1.4.1 also requires organizing and presenting data, which fits with recording the combinations found in a clear way.

Typical mistake: Anyone who writes down combinations without a fixed order either counts one possibility twice or overlooks one. If, for example, one type of bread is paired with a topping, it is easy to lose track of which pairs have already been formed. A fixed order, such as always listing all pairs with the first type first and then all pairs with the second type, makes this happen less often.

Typical tasks in this area are:

• How many different pairs: true or false
• How many different combinations are possible
• Write down all combinations
• How many combinations from three types
• How many pairs remain if one part breaks
• How many more pairs after an additional part
• Known number of pairs: how many in the other set
• How many different pairs: choose the correct answer

The worksheets for these tasks contain solutions. Probabilities are not practiced separately in the tasks available here.

The child can read information from a table or a chart and make simple comparisons with it. The data comes from everyday life, and the representations are tables, column charts and bar charts. The curriculum points on which this text is based do not specify a concrete numerical range for this.

A typical mistake when comparing two rows or bars is reading the wrong place, for example because one looks for the name instead of the numerical value, or judges the height of a bar by eye instead of by the value on the axis. When asked about a difference, the order of subtraction is also often mixed up. Anyone who reads the question only halfway then calculates "how many fewer" instead of "how many more". Reading the question carefully usually helps more here than calculating quickly.

In practice, tasks on this topic look like this:

• How many more according to the table: compare two rows and state the difference.
• Difference between two rows: either select the correct answer or judge a statement as true or false.
• The gap between the largest and the smallest: find the largest and the smallest value and calculate the difference.
• Column chart: how many more: the same comparison question, this time using the chart.
• Which category has the most: recognise the highest value and name the category.
• How many groups lie above the given number: count bars or rows whose value is greater than a specified number.

The curriculum describes working with data over a longer period of time, including recognising manipulations in statistical representations. The tasks at this grade level do not yet address this; they are limited to reading and comparing.

The child learns to calculate the perimeter of a rectangle. The perimeter is the length of the edge, that is, the sum of all four sides: twice the length and twice the width. The curriculum points do not name a specific number range, so the principle is the focus here rather than concrete numbers.

A typical mistake is to add only the length and the width and forget the other two sides. Another common mistake is multiplying the sides. This happens because multiplication belongs to area, whereas perimeter is a path along the edge. On squared paper, children often count the squares inside instead of the squares along the edge.

The tasks in this class look, for example, like this:

• Perimeter of a rectangle: The sides are added together.
• Missing side of a rectangle: The unknown side is calculated from a given perimeter.
• Which figure has the greater perimeter: Two figures are compared.
• Perimeter of a rectangle on squared paper: The length of the edge is counted.
• Is this perimeter correct: A given solution is checked.
• Draw a rectangle with the given perimeter: A suitable rectangle is constructed.
• Choose the correct perimeter: The correct one is selected from several options.

The child can, for tasks with two calculation steps, such as 3 + 4 · 2 or 5 · 6 − 7, first work out the correct step and then the second. The curriculum point names arithmetic operations with natural numbers and non-negative decimal numbers. However, the tasks in this area for this class work with whole numbers in real-life situations. The order of operations itself is not explicitly named in the curriculum point but is practised in the tasks.

A typical mistake is to simply calculate the task from left to right. From 3 + 4 · 2, this gives 14 instead of 11, because the child first adds the 3 and the 4. The reason is often that the child does not recognise which step is meant to come first, or that the task is read only as a sequence of numbers and not as a situation.

In practice, a task might look like this: four boxes each contain six apples, and five single apples are added. The child first works out how many apples are in the boxes, and then how many there are altogether. The templates for this area include, among others:

• Order of arithmetic operations (two steps)
• First how many in one, then how many altogether
• So many groups of so many each, then some part is taken away
• Two kinds of groups, one sum
• How many remain when everyone gets the same amount
• Order of operations: choosing the correct result
• Order of operations: right or wrong

Your child can compare which outcome is more likely in simple chance experiments, for example when a ball is drawn blindly from a bag. The curriculum points describe this comparing qualitatively, that is, without calculating numbers. The curriculum also mentions determining probabilities, which builds on an intuitive understanding. However, the tasks in this grade mainly focus on comparing.

Often only the number of colours is considered: a bag with one red and one blue ball seems "fair," even though with three red balls and one blue ball there is clearly more red. Likewise, the number of red balls alone is often compared without paying attention to the total number. Two red balls out of two balls is a very different chance than two red balls out of ten. The reason is that what matters is how many balls there are in total.

Tasks on this topic take the form shown by the templates in this area:

• Which chance is greater? Your child chooses between two events.
• Fill the bag: Your child places balls in the bag so that one chance is greater, smaller or equal.
• Improve the comparison of chances: An incorrect statement is corrected.
• Did the character compare the chances correctly? The reasoning of a character is checked.
• What are you most likely to draw? and A bag with balls: what is more likely? Your child decides which outcome is most likely.

The child reads tables and picture graphs taken from everyday life and extracts information from them: how many entries there are in total, which value is missing in a row, or whether a statement about the total is correct. The curriculum points on which this text is based describe collecting and displaying data in tally lists, tables, column and bar charts, and pictograms across several school years. They do not name a specific number range, so the level of this class shows mainly in the tasks themselves.

A typical mistake with picture graphs is counting only the pictures, even though one picture stands for several units. If one symbol means five and three symbols are shown, the value is 15, not 3. With the missing row, the known total is often used incorrectly: the other values are added instead of being subtracted from the total. For statements that are true or false, it helps to actually recalculate the table rather than decide by feeling.

The templates in this area require reading and completing; drawing your own diagrams is not one of the printed task formats. Typical tasks are:

• How many in total in the table: add up all the entries.
• Picture graph: one picture stands for several: calculate the value from the number of pictures and the value of one picture.
• The missing row with a known total: determine the missing value from the total and the other rows.
• Total from the table: choose the correct answer and Total from the table: true or false: check statements about the total.

The curriculum describes for this grade level that arithmetic operations with natural numbers and with non-negative decimal numbers should be carried out and interpreted, and additionally estimation calculations. This point applies to several school years and does not name a separate number range for the first grade. Specifically practiced in this class is adding and subtracting decimal numbers, that is, calculations such as 2.5 + 0.75 or 5 − 0.8.

A typical mistake arises when the numbers are not written with the decimal points aligned, but so that the last digits are stacked on top of each other. Then tenths are combined with hundredths, and the result is wrong. In subtraction, the idea that you can write 5 as 5.0 before subtracting 0.8 is often missing.

Tasks on the worksheets for this class are, for example:

• Adding decimal numbers: adding two decimal numbers together in written form.
• Subtracting decimal numbers: subtracting one decimal number from another.
• Adding masses with decimal numbers: adding weights or lengths such as 1.25 kg and 0.4 kg and interpreting the result in a real-world context.

An estimate, in which you round the numbers roughly beforehand, helps to check the result. According to the curriculum point, it belongs to the requirements, but it is not listed as a separate task type on the worksheets.

In this class, children calculate with non-negative decimal numbers such as 3.5 or 0.42 and interpret what the calculation means. The curriculum point does not specify a fixed number range. The types of tasks make clear what is required: multiplying decimal numbers by 10, 100 and 1000, dividing them by 10, 100 and 1000, and multiplying them by 0.1, 0.01 and 0.001.

A typical mistake is applying the rule "to multiply by 10, add a zero" to decimal numbers. For 4.5 · 10, this gives 4.50 instead of 45. The correct approach is that every digit moves one place to the left, so the 4 moves to the tens place and the 5 moves to the ones place. Many children also confuse "multiplying by 0.1" with "multiplying by 10", even though it is the same as dividing by 10. An estimate helps as a check: 0.38 · 100 gives 38, not 3.8 and not 380.

An example task looks like this: 7.25 · 100 or 96.4 : 10. For tasks with 0.1, 0.01 and 0.001, the result becomes smaller. The printable worksheets contain solutions so that parents can check the results. As an additional practice idea, children can estimate for each task first whether the number must become larger or smaller.

Your child learns to describe the properties of cuboids and to justify and apply the formula for their volume. The curriculum point does not name a specific number range, so none is given here. What matters is the understanding behind it: a cuboid can be filled with layers of unit cubes, where length times width gives the number of cubes in one layer, and height gives the number of layers.

A common mistake is to add the three edge lengths or to multiply only two of them. This often stems from confusing it with surface area, which measures the outer surface, or the child has forgotten the layers when counting. Mixed units, such as centimetres and millimetres within one task, also easily lead to incorrect results.

Tasks in this area typically look like this:

• Volume of a cuboid: The edge lengths are given, and the child calculates the volume.
• Volume from unit cubes: The child counts cubes in layers or multiplies the number of layers by the number of cubes per layer.
• Volume of a solid made of two cuboids: The solid is divided into two cuboids, whose volumes are added together.

Your child learns to convert lengths between units and to write a length both in a single unit (2.4 m) and in multiple units (2 m 40 cm). The curriculum names the length units mm, cm, dm, m and km for this. The curriculum does not specify a separate number range for this part.

A typical mistake occurs when switching between decimal notation and multi-unit notation. If 3.5 m is written as "3 m 5 cm", the child reads the digit after the decimal point as a whole number of the smaller unit. The correct form is 3 m 50 cm, because 0.5 m is 50 cm. Another common error is confusing the direction: when converting to a smaller unit, the numerical value gets larger (2 m = 200 cm); when converting to a larger unit, it gets smaller.

Such tasks come in three forms:

• Converting to a smaller length unit, e.g. 4 m in cm
• Converting to a larger length unit, e.g. 750 m in km
• Expressing a length in one unit, e.g. 2 m 40 cm as 2.4 m

The primary school curricula also require estimating lengths. An image such as "a classroom door is about 2 m high" helps to check whether a converted number is plausible. The example tasks are available as printable worksheets with solutions.

The mean is the average of several numbers: you add all the values and divide the sum by the number of values. The curriculum points describe a longer period and do not name a specific number range. However, the tasks in this area show what this class is about: the child should calculate means and understand what they tell us.

A typical mistake is dividing by the wrong number, for example when one value is overlooked while counting. Children also often confuse the mean with one of the values that occurs or with the middle value of an ordered series. With bar charts, there is the added step that the values must first be read correctly before any calculation can be done.

Suitable tasks in this area are:

• Calculating the mean: Add several numbers and divide the sum by the number of values.
• Mean from a bar chart: Read the values from the bars and determine the mean from them.
• The mean in everyday life: Work with values from daily life and explain the result in words.

In this class, the child calculates the area of a rectangle by multiplying length and width. The child can also work in reverse: if the area and one side are given, they determine the other side. The curriculum points describe a longer stretch of schooling and do not name a number range for this class. Which numbers actually appear is therefore shown by the tasks on the worksheets.

Typical errors usually have two causes:

• Perimeter and area are confused. The child adds the sides although the area is asked for, or multiplies although the perimeter is sought.
• The missing side is calculated incorrectly. The child multiplies the area by the known side, although they would need to divide. A helpful check is: the known side times the side found must give the area again.

Tasks of the type Area of a rectangle require direct calculation. Area of a rectangle in everyday life places the same calculation in real-life situations, such as a room or a garden bed. In Missing rectangle side, the area and one side are given and the other side is sought. All worksheets include solutions. As a personal aid, the child can draw a rectangle on squared paper and count the squares; this is a suggestion for support and not a mandatory requirement.

The child learns to substitute a number for the letter in a term or formula and to calculate the resulting value. The curriculum requires setting up and interpreting simple terms, equations and formulas. It does not name a number range, so this page only describes what the task templates in this area show.

A typical mistake is not reading a number placed directly next to a letter as multiplication. For 3 · a with a = 4, one then calculates 3 + 4 or writes "34". With formulas, the order of operations is also sometimes ignored, for example when one calculates from left to right even though multiplication comes before addition and subtraction.

The task templates in this area are:

• Value of a term with a variable: A given number is substituted for the letter, and the term is calculated.
• Value of a formula: With given values, a formula is calculated, for example for an area or a perimeter.
• How many times as large? Without calculating: The child compares two quantities without fully calculating them.

The curriculum point requires interpreting, representing, and comparing natural numbers. Roman numerals are one of several ways to represent natural numbers. The child learns to read a number written in Roman letters as a number, and conversely to write a number in this notation. This topic covers Roman numerals up to 3000. However, the curriculum point itself does not name a number range for the Roman notation.

A typical mistake arises with the subtraction rule. When a smaller digit stands before a larger one, you subtract: IV means 4, not 6. Anyone who simply adds the symbols together reads IV as 6, IX as 11, or XL as 60. When writing, the opposite mistake happens: the number 40 is written as XXXX instead of XL, because the short form is not known.

A task on this topic usually looks like this:

• Read a Roman numeral: A given Roman numeral, such as MCMXC, is stated as a natural number.
• Write a number in Roman numerals: A given natural number is written in Roman numerals.

Both tasks are available as printable worksheets with solutions. This makes it possible to check whether the subtraction cases were read and written correctly.

The curriculum states that non-negative fractions should be interpreted, represented, and compared. The exercises on this topic practise this mainly through converting between improper fractions and mixed numbers, in both directions.

A typical mistake occurs with mixed numbers. From 2 3/4 one gets 5/4, because 2 and 3 are added and the denominator 4 is kept. The correct approach: 2 wholes are 8 quarters, plus 3 quarters makes 11/4. Conversely, 11/4 is sometimes written as 2 4/3, because numerator and denominator are swapped. The correct approach: 11 divided by 4 gives 2 remainder 3, so 2 3/4.

The printable worksheets mainly contain these types of exercises:

• Converting an improper fraction into a mixed number, for example 7/3 = 2 1/3
• Converting a mixed number into a fraction, for example 3 2/5 = 17/5

As a personal aid, not as part of the exercise templates, the numbers can be visualised with a drawing, for example with rectangles divided into equally sized parts.

The curriculum item requires performing and interpreting arithmetic operations with non-negative decimal numbers, as well as making estimates. It does not name a separate number range. In this grade, rounding is practiced concretely to two places: to tenths and to hundredths.

A typical mistake happens with carrying over. The child correctly looks at the digit to the right of the place being sought, but then changes the digit before it incorrectly. Rounding 2.96 to tenths gives 3.0, not 2.6. It is also often forgotten that the number after rounding is still a result with the desired number of decimal places, so 4.5 rounded to hundredths is written as 4.50.

A task looks like this, for example:

• Round to tenths: 3.47 becomes 3.5.
• Round to hundredths: 0.826 becomes 0.83.

The printable worksheets with solutions are suitable for practice. The rounded result can also be checked with a quick estimate, for example by considering whether the answer is close to the original number.

According to the curriculum, children should carry out arithmetic operations with natural numbers and with non-negative decimal numbers, interpret them, and perform estimation calculations. For multiplication, this means: your child calculates with decimal numbers, understands what the result means, and checks with a rough calculation whether the value is plausible. The curriculum does not specify a separate number range here, so we only describe what the tasks on the worksheets show.

A typical mistake: the child calculates as if there were no decimal comma and places the comma in the wrong position at the end. Results such as 0.1 instead of 10 for 2.5 times 4 often occur because the digits after the decimal point are not counted. Estimation helps here: 2.5 times 4 is roughly 2 times 4, so about 8. A result of 0.1 or 100 will then stand out immediately.

In the foundation-level volume for Grade 1, there are two task types on the printable worksheets:

• Multiplying decimal numbers: Two numbers with decimal commas are multiplied together, and the result is checked with an estimation calculation.
• Multiplying a length by a quantity: A ribbon is 2.5 m long, and 4 ribbons are needed. The total length is 2.5 m times 4 = 10 m. This shows how the result is interpreted: the unit is retained and refers to the object.

Your child learns to translate a verbal description into a mathematical expression with letters. For example, "4 more than three times a number x" becomes 3 · x + 4. The curriculum requires setting up and interpreting simple terms, equations and formulas. The curriculum item does not specify a number range.

A typical mistake is taking the order of the words directly into the calculation. Then the numbers get swapped, for example 4 · x + 3 instead of 3 · x + 4, or the factor "three times" is forgotten and only x + 4 is written. The reason: the sentence describes a meaning, and the order of the words does not always match the order of the calculation steps.

In this math area there are two task types for this skill:

• Set up a term: more than a multiple. Your child writes the matching term for a description with one variable.
• Set up a term with two variables. Two letters appear here, for example for two different quantities, and the term connects them.

When checking, it helps to substitute a number for the letters and verify whether the term reflects the description.

The child can determine the perimeter of a polygon by adding up all its side lengths. The curriculum also requires calculating the perimeter of squares and rectangles. This point is in the primary school curriculum, and the tasks on this page show the part that is printed in the 1st grade of Mittelschule or AHS. A specific number range is not named in the curriculum for this point.

A typical mistake arises with the missing side: many children add up the known sides and the perimeter, instead of subtracting the known sides from the perimeter. Often a side is also forgotten when adding, especially with shapes that have more than four corners, because not all sides look the same.

The template Perimeter of a polygon requires adding up all the side lengths of a figure. The template Missing side from the perimeter gives the perimeter and all but one side; the child first adds the known sides together and subtracts this sum from the perimeter. The worksheets on this topic page are printable and include solutions for checking.

The child works with simple geometric figures in the plane. Here the focus is specifically on the circle: if the radius is given, the child determines the diameter, and if the diameter is given, the child determines the radius. The curriculum points do not name a number range for this, so this text only describes the types of tasks that occur in this area.

Children often confuse the two terms or forget the factor 2. The radius runs from the center to the edge, while the diameter goes through the center from edge to edge. The diameter is therefore twice as long as the radius. If a child simply takes the radius over as the diameter, or halves when they should be doubling during the conversion, this confusion is usually the cause.

Typical tasks on the worksheets are:

• Diameter from the radius: The radius is given, the diameter is to be found.
• Radius from the diameter: The diameter is given, the radius is to be found.

Both task types are available as printable worksheets with solutions, so you can check the calculation together with the child.

The child translates everyday situations involving money into calculations. They write amounts in euros and cents (for example 2 € 35 ct or 2.35 €) and interpret price information correctly. If a table shows that doubling the number of items doubles the price, the child can recognise this directly proportional relationship and use it for calculations. The curriculum points do not name a number range for this grade. We therefore describe here only what the task templates show.

A typical mistake arises when reading decimal numbers. Many children treat 2.5 € as "2 euros 5 cents" instead of "2 euros 50 cents". Then 0.8 € seems larger than 0.75 €, because 8 is larger than 75. A second common mistake: two offers with different quantities (for example 500 g and 1 kg) are compared directly without converting the units of measurement. Both are related to working with decimal numbers and unit conversions.

In practice, the tasks look like this:

• Which offer is cheaper? The child compares two prices and justifies their decision with a calculation.
• How much money is left over? The child subtracts the expenses from the starting amount and can also check whether the change is correct.

As a helpful strategy of their own, not a curriculum requirement, when comparing one can first relate both prices to the same quantity, for example to 1 kg.

The Volksschule curriculum states that numbers up to 1,000,000 are to be read, written, represented, decomposed, compared, ordered and rounded. This area extends over several school years. In Grade 5, according to the task templates for this area, the focus is mainly on comparing large numbers and understanding place values.

A typical mistake is reading numbers digit by digit instead of in groups of three, such as "482 905" (482 thousand 905). When writing, zeros are forgotten, so that "two million three hundred" becomes a number like 2 300. When comparing, some children look only at the last digit or the first digit. The correct approach is: first count the number of digits, then look at the leftmost digit where the numbers differ. The cause usually lies in the fact that the meaning of a digit depending on its place value is not yet securely anchored.

In the worksheet "Comparing large numbers," the child places the symbols <, > or = between two numbers, for example 482 905 and 489 210. In "How many times as large? Place values," it is determined how many times one number is greater than another, for example 3 000 compared with 300 (ten times as large). All tasks are available as printable worksheets with solutions.

Your child learns to set up an equation with one variable from a word problem, solve it, and relate the result back to the situation. The curriculum points describe this in general terms for terms, equations and formulas in various situations. They do not explicitly name a specific number range for this grade level.

A typical mistake lies in translating words into arithmetic operations. For "5 less than the number", 5 − x is often written instead of x − 5. In tasks where the starting quantity is sought, the stated changes are sometimes calculated in the wrong direction: what was added is added, although it should be subtracted back. The reason is usually that the sought quantity was not set up as a variable, but calculated directly with numbers.

The task templates in this area come in two forms:

• What number was it? The text describes an unknown number, for example through a calculation involving it. The child sets the number as x, writes the equation and solves it.
• How many were there at the start? The text describes what happened and gives the final amount. The starting quantity is the unknown that appears in the equation.

The worksheets contain solutions with which you can compare the results. A helpful step is to insert the calculated solution into the original text and check whether it fits the story. The curriculum points also mention fractions, proportionality and percentages; the templates available here do not show separate examples for these.

Your child can calculate terms with brackets. In this case, the part inside the brackets is calculated first, and then the result is substituted into the rest of the term. The curriculum names natural numbers and non-negative decimal numbers for this. It does not name a separate number range for bracket terms. The curriculum points also describe a longer section of schooling and not just this class.

A typical mistake is to overlook the brackets and simply calculate the term from left to right. For 3 · (4 + 2), this gives 3 · 4 + 2 = 14, but the correct answer is 3 · 6 = 18. The mistake happens because the brackets are seen only as a way of writing and not as an instruction to calculate first. A quick estimate helps: if the result is clearly smaller than expected, the brackets were probably ignored.

Tasks on this topic look like a term with a bracket is given and must be calculated. The template for this is called "Calculate term with brackets". For example:

• (7 + 5) · 3
• 20 − (6 + 4)
• (2,5 + 1,5) · 2

The worksheets with solutions allow you to check the results yourself.

When estimating, the numbers in a calculation are rounded to simple values so that the approximate result can be worked out in your head. This estimate serves as a check: if the exact calculation differs strongly from it, it is worth checking again. The primary school curricula relate estimation to the number ranges 1000 and 1,000,000. For this grade, the curriculum specifically states: carry out estimation calculations with natural numbers and non-negative decimal numbers.

A typical mistake occurs with decimal numbers: instead of rounding to the nearest whole number, the digits after the decimal point are simply dropped. For example, 4.8 is counted as 4, even though 5 is the closer number. With several addends, such deviations add up, and the estimate ends up noticeably off. It helps to ask for each number: is it closer to the whole number below or the whole number above?

The worksheets on this topic include the task type estimate the sum. For example, for 3.8 + 6.4 + 2.9, you first calculate with rounded whole numbers (4 + 6 + 3 = 13) and then carry out the exact calculation (13.1). The solutions on the sheet help you compare the estimate and the exact result with each other.

Your child is learning to expand fractions: the numerator and the denominator are both multiplied by the same number. The fraction then has a larger denominator, but its value stays the same. For the number domain, the curriculum sets a goal for the whole grade level, namely to interpret, represent and compare natural numbers as well as non-negative decimal and fractional numbers. This worksheet offering for Grade 1 focuses mainly on expanding fractions.

A typical mistake is to multiply only the denominator or only the numerator, for example 1/2 = 1/4 instead of 2/4. Just as often, the same number is added instead of multiplied, for example 1/2 = 2/3. This often stems from the idea that the numerator and denominator are two separate numbers rather than a part-of-a-whole relationship. A drawing helps: if a circle or a strip is divided into equal parts, it is easy to check whether the proportion has stayed the same.

A task might look like this: "Expand 3/5 by 2" or "Fill in: 3/4 = ?/12". The worksheets on expanding fractions contain example tasks of this kind, and the solutions help you check your work.

The child learns to classify angles by their size. The following types are mainly distinguished:

• acute: smaller than a right angle
• right: exactly as large as a corner of a square
• obtuse: larger than a right angle, but smaller than a straight angle

The curriculum assigns this topic to working with simple geometric objects in the plane. At this point it does not name a specific numerical range.

A typical mistake is to believe that the length of the arms determines the size of the angle. An angle with long arms can be acute, and an angle with short arms can be obtuse. Only the opening between the two arms is decisive. A second mistake occurs with angles that are just above 90°: they are easily called right, because the deviation is hard to see with the eye.

A suitable task can be found in the worksheet collection under the title Die Winkelart bestimmen (Determining the Angle Type). The child sees several angles and assigns the appropriate type to each one. This shows whether the child pays attention to the opening and not to the length of the arms.

The curriculum point requires interpreting, representing and comparing natural numbers as well as non-negative decimal and fraction numbers. It describes a multi-year period and does not name its own number range for grade 1. For this grade, the existing task templates show what is concretely practised: writing a quotient of two natural numbers as a fraction.

A common mistake is to swap dividend and divisor, i.e. to write 3 : 4 as 4/3. This often happens because the order in the division sign is not consciously carried over to the fraction. The following rule helps:

• The number before the colon becomes the numerator (top).
• The number after the colon becomes the denominator (bottom).

A typical task is, for example: Write 5 : 8 as a fraction. The solution is 5/8. Such tasks can be found as the template "Write a quotient as a fraction" on the printable worksheets, each with solutions.

Your child can write a decimal number as a fraction, for example 0.75 = 75/100. This belongs to the curriculum point of interpreting, representing and comparing non-negative decimal and fraction numbers. The curriculum describes this point for a whole learning stage. For this grade, what counts is what the tasks show: decimal numbers are converted into fractions.

A typical mistake is to simply take over the digits and overlook the place value. Then, for example, 0.5 is written as 1/5 or 0.25 as 25/10. The reason: the child orients themselves on the digits instead of the last place after the decimal point.

• One place after the decimal point means denominator 10 (0.5 = 5/10).
• Two places mean denominator 100 (0.25 = 25/100).
• Three places mean denominator 1000 (0.125 = 125/1000).

The printable worksheets on "Writing a decimal number as a fraction" give decimal numbers that should be written as fractions. The sheets include solutions for self-checking. At home, it helps to count together how many places follow the decimal point before the denominator is determined.

The curriculum for mathematics states that students should interpret, represent and compare non-negative decimal and fraction numbers. Rewriting a fraction as a decimal is part of this: a fraction such as ½ and the decimal 0.5 denote the same quantity. The curriculum extract does not name a specific number range, so none is given here.

A typical mistake occurs when the digits of the denominator are simply taken over as decimal places. From ¼ one then gets 0.4 instead of 0.25, because only the 4 is copied and no thought is given to which part of the whole is meant. A helpful question is: does the decimal have the same value as the fraction?

On the printable worksheets, this skill appears in the template Writing a fraction as a decimal. Here, fractions are given that the child should convert into decimal notation. Each task comes with a solution, so the result can be checked.

The curriculum requires that students interpret, represent and compare non-negative decimal and fraction numbers. This curriculum point applies to the entire lower secondary level and not only to the 1st grade. Which number of decimal places is specifically expected in this grade cannot be determined from the information available. Comparing includes understanding what the digits after the decimal point mean and being able to decide which of two numbers is larger.

A typical mistake is treating decimal places like whole numbers. Then 0.45 appears larger than 0.5, because 45 is larger than 5. The error arises because the child does not take place value into account: the 5 in 0.5 stands for five tenths, while the 45 in 0.45 stands for four tenths and five hundredths.

The worksheets on this topic include the task Comparing decimal numbers. In it, your child places the appropriate symbol (<, > or =) between two decimal numbers. As a practical tip from experience, which is not part of the curriculum, you can append the same number of decimal places to both numbers (0.50 and 0.45), so that the digits can be compared directly next to each other.

The child should carry out and interpret arithmetic operations with non-negative decimal numbers and make estimation calculations. Curriculum point U1.1.2 applies to a multi-year section and does not name a specific number range. What is actually printed in Grade 1 is shown by the task template "Dividing decimal numbers".

A typical mistake is to treat the decimal comma when dividing as if it were a whole number. The position of the comma in the result is then set incorrectly, and the result has the wrong order of magnitude. Estimation helps here: anyone who estimates 12.6 : 3 (12 : 3 is about 4) will immediately notice if 0.42 or 42 comes out.

The printable worksheets include tasks based on this template, each with solutions. An example for illustration would be a division task with a comma in the result. The child can proceed as follows:

• Estimate beforehand roughly what size the result should have.
• Calculate the task exactly.
• Compare the result with the estimate and check the position of the decimal comma.

The curriculum states that children should describe the properties of cuboids and justify and apply formulas for surface area and volume. This includes understanding that a cuboid has six faces, which are equal in pairs, and that surface area is understood as the sum of these faces. The curriculum does not name a specific number range for this topic; the exercises show which numbers come up in class.

Children often make two different mistakes here. Either only some of the faces are counted, for example just the base and the top, or the length and width are multiplied without including the remaining faces. Another typical mistake is confusing surface area with volume: surface area is given in square units, while volume is given in cubic units. It helps to actually draw the faces or to look at a net of the cuboid.

A typical task is the worksheet "Surface area of a cuboid." In it, the length, width and height of a cuboid are given, and the child calculates the surface area. The worksheets are printable and include solutions, so parents can easily follow the results.

The child writes a measurement given in two units, such as 2 m 5 cm, as a single decimal number in one unit, here 2.05 m. In doing so, the child reads the measurement correctly: it recognizes which part is given in metres and which in centimetres. The curriculum does not name a separate number range for this point.

A typical mistake is to write 2 m 5 cm as 2.5 m. The reason lies in place value: one centimetre is the hundredth part of a metre, so 5 cm = 0.05 m. The centimetres therefore need two places after the decimal point, and the digit 5 stands in the second place, not the first.

The tasks in this area are titled "Expressing length as a decimal number". There are printable worksheets with solutions, for example:

• Write 3 m 40 cm as a decimal number in metres. (Solution: 3.40 m)
• Write 1 m 8 cm as a decimal number in metres. (Solution: 1.08 m)

The curriculum guidelines specify litres (ℓ) and millilitres (mℓ) as units for quantities. Your child should understand measurement units, write quantities with one or several units, and carry out conversions. The curriculum points do not specify a separate number range for this, so here we only describe what is practised in this grade: converting into a smaller volume unit.

A typical mistake comes from expecting that a smaller unit also gives a smaller number. Someone who converts 2 ℓ into millilitres sometimes writes 2 mℓ or calculates with 100 instead of 1000, because that is familiar from converting centimetres and metres. The correct answer is: 1 ℓ = 1000 mℓ, so the number becomes larger when changing to the smaller unit, namely 2 ℓ = 2000 mℓ.

A task on this might look like this: “How many millilitres are 3 ℓ?” or “Write 2500 mℓ as litres and millilitres.” You will find such tasks in the section “Converting into a smaller volume unit” on the printable worksheets with solutions. A helpful way to picture it (our own suggestion, not part of the curriculum requirements): a milk carton holds about 1 ℓ, that is, 1000 mℓ.

The child learns to multiply two multi-digit numbers in writing. The primary school curriculum names as its goal multiplication with a single-digit and a two-digit multiplier in the number range up to 1,000,000, where results should be estimated by rounding and approximate calculation. The curriculum for Mittelschule and AHS lists arithmetic operations with natural numbers and non-negative decimal numbers, likewise with estimation calculations.

A typical mistake arises with a two-digit multiplier: the second partial product is not shifted one place to the left. For example, with 24 · 13, 24 · 3 = 72 is calculated correctly, but then 24 · 1 = 24 is written directly underneath instead of 240. The sum then gives 96 instead of 312. A quick estimate helps to notice such errors: 24 · 13 is about 25 · 12 = 300.

On the printable worksheets on the topic "Multiplying multi-digit numbers in writing," children practise this procedure with multi-digit numbers. The solutions are included, so parents can check the calculations together with the child.

Your child can read and compare values from a bar chart and from a table: Which bar is the tallest? Which values are equal? How big is the difference? The curriculum requires reading data from different forms of representation. For this topic, the guidelines do not name a separate number range.

A typical mistake is that the child judges the bars only by their height and overlooks the scale at the side. If one unit covers several squares or the scale does not start at zero, bars of equal height can show different values, and appearance is misleading. The error arises because a bar easily looks like a quantity, even though only the scale fixes its value. As a suggestion of our own for practice: before each answer, the child says aloud what the scale shows for the bar.

A task on this topic typically involves comparing bars in a chart. The child first reads the value of each bar and then answers questions such as:

• Which bar is the tallest?
• Which bars show the same value?
• How much greater is one value than the other?

The curriculum also mentions relative frequencies and recognising manipulations in representations. However, the task examples for this class only show comparing bars; exercises going beyond that are not included in the tasks offered here.

In the 1st grade of Mittelschule or AHS, the child learns to interpret, represent, and compare non-negative decimal numbers. The curriculum point does not specify a fixed number of decimal places. The tasks in this area show, for this grade above all, reading off the hundredths digit: in 3.47, the 4 is in the tenths place and the 7 is in the hundredths place.

A typical mistake arises when the child treats the digits after the decimal point like whole numbers. Then, for example, in 0.38 the 3 is read as hundredths because it stands at the front, or the place value is determined only by counting the digits. It also often happens that 0.25 seems larger than 0.4 because 25 is larger than 4. The reason: the value of a digit depends on its position after the decimal point, not on the digit itself.

A task on this topic looks like this: the child is shown a decimal number and must name the digit in the hundredths place. The worksheets in this area contain the template "Reading the hundredths digit", with solutions for checking. One possible help at home (my own suggestion, not a curriculum point): label the places from left to right with tenths, hundredths, and so on, and write the digits one by one under the labels.

Your child learns to describe, for two line segments in the plane, whether they run parallel or stand perpendicular to each other. The curriculum does not specify a number range; what matters is the position of the segments relative to one another. The same curriculum item also covers drawing right angles with a set square.

Often only the visual impression is used: a segment lying diagonally on the page is then considered "somehow different", even though it may be parallel to another. The terms parallel and perpendicular are also easily confused. It helps to count the segment on the grid: how many squares does it go horizontally, and how many vertically? This is a learning aid from us and not a separate curriculum item.

In the task "Parallel or perpendicular?", the child sees segments on a grid and decides for each pair whether it is parallel or perpendicular. The worksheets are printable, and there is a solution for each task for comparison.

The curriculum item requires working with simple geometric objects in the plane. Measuring angles with a set square (Geodreieck) belongs to this area. The curriculum does not name a specific numerical range, so here we only describe what the tasks in this class show.

A common mistake is reading the wrong scale. The Geodreieck has two opposing angle scales, and both start at 0. If the zero mark does not lie exactly on one arm of the angle, the result is wrong. Then the angle is often drawn correctly, but the number is read from the wrong scale, and the result deviates considerably.

In the worksheets, the task Reading angles on the Geodreieck shows how the child places the Geodreieck at the vertex and reads off the degree value. A helpful check is to estimate beforehand whether the angle is acute, right or obtuse, and to compare the result with that estimate.

The child distinguishes drawn lines by how they are bounded. A line segment has two endpoints. A ray has only one starting point and continues infinitely in one direction. A line has no beginning and no end and continues infinitely in both directions. The curriculum points describe a period of several school years. This page describes only the part that is practiced in grade 1 using the task templates.

A typical mistake is confusing a ray and a line segment, because both have a visible starting point. The line is also often called a "particularly long line segment." This is because on paper every line is only an excerpt. The distinction becomes visible only through the arrowheads and the question of whether and where the line ends:

• Two endpoints, no arrows: line segment.
• One starting point, an arrow at the other end: ray.
• Arrows at both ends: line.

The matching task template for this is called "Line segment, ray or line?" In it, the child assigns drawings to one of these three terms and justifies the choice with endpoints and arrows. The curriculum point also includes drawing right angles, squares and rectangles with a set square. However, the available tasks on this topic only show the assigning. A separate template for drawing with a set square does not exist here.

The curriculum describes axially symmetric figures across several school years: children should recognize them, construct them, and use their properties. In this class, the worksheets mainly cover one part of this, namely reflecting individual points across a straight line.

The child draws the image point for a given point and a given line (the axis of symmetry). The image point lies on the other side of the line, at the same distance from it and perpendicularly opposite the starting point. For these tasks, the curriculum does not specify a separate number range.

Typical errors and their causes:

• The image point lies on the same side as the starting point. This happens when the point is shifted instead of reflected.
• The distance is measured at an angle or by eye instead of perpendicular to the line. As a result, the image point ends up too far away or too close to the axis.

A task from the template "Reflect a point across a line" gives a line and a point; the child completes the image point. As a check, the sheet can be folded along the line: if the point and the image point coincide, the solution is correct. This tip is a suggestion and not part of the original instructions.

The child examines plane figures and solids, describes their properties and compares them, including through play, for example with everyday objects that are symmetrical. The curriculum points do not name a number range for this, because it concerns properties and not quantities. Typical features are:

• Prism: two equally large, parallel base faces, and the remaining faces are rectangles (for example a triangular prism or a cuboid).
• Pyramid: one base face and triangular faces that meet at a single apex.
• Cylinder: two equally large circular faces at the top and bottom and one curved lateral surface.
• Cone: one circular base face and one apex, with no edges.
• Sphere: round, with no vertices, edges or flat faces.

A common mistake arises when children focus only on the apex or on the shape of a single part. A cone is then confused with a pyramid because both have an apex, or a lying cylinder is taken for a cone. A cuboid is also often not recognised as a prism, because children think only of triangular prisms. It helps to count and name the faces instead of deciding based on the first impression.

One task on this topic is, for example, the worksheet "Which solid is this?" The child identifies a pictured solid by its faces, edges and apexes and names it. The curriculum points also describe calculating the surface area and volume of right prisms, as well as the volume of pyramids. Such calculations are not offered in the exercises in this area; here the focus is on recognising and describing.

The child can measure or estimate the area of flat figures using unit areas and calculate the area of squares and rectangles. For simple figures composed of rectangles, the child divides the figure into partial rectangles, calculates their areas, and adds them together. The curriculum does not specify a number range for this point.

A side length that is not directly given is often missing. The child then calculates with the total length instead of the remaining length, which results from the difference. Another typical mistake is confusing areas with perimeters or writing the unit cm instead of cm².

The worksheets for this class focus mainly on the task area of an L-shaped figure. One possible approach is:

• determine missing side lengths from the given measurements,
• divide the figure into two rectangles,
• calculate the two areas and add them,
• state the result with the correct unit.

The solutions are included on the worksheets.

In the area of body models, the child learns to examine, match, and create nets of simple solids. The focus is on shapes and arrangements, not on numbers. The curriculum point U1.3.3 on cuboids (justifying properties, surface area, and volume) is a goal across the whole stage. On this page, for grade 5, only the cube net is practised for nets.

A typical mistake: the child only counts whether there are six squares and considers every such figure a cube net. When folding, faces then overlap, or one side of the cube remains open. Two examples that are not cube nets: a rectangle made of two rows of three squares, where all the squares lie exactly on top of each other, or four squares in a row with two adjacent squares on one side. Often the child looks at appearance ("looks like a cube") instead of asking whether every face of the cube occurs exactly once.

A task looks like this: the figure made of squares is shown, and the child decides with "Yes" or "No" whether it is a cube net and justifies the answer. The template for this is "Is this a cube net?". As an own idea for practice (not as a requirement): draw the figure on cardboard, cut it out, and fold it to check the decision.

The child learns to solve simple equations: a number in a calculation is unknown, and the child finds it by working through the calculation backwards. The curriculum point does not name a number range, so none is given here either.

A typical mistake is to use the same operation as in the task when solving it. Anyone who sees "plus 3" and adds 3 again gets a wrong number. Often the steps are also undone in the wrong order. The reason: children usually calculate forwards, and the question about the starting number requires reversing the direction and solving the last step first.

For this area there is the task template Thought-of Number. An example: "I think of a number, double it and add 3. The result is 17. What number did I think of?" The child works through the steps backwards:

• 17 − 3 = 14 (the addition is undone)
• 14 : 2 = 7 (the doubling is undone)